Abhyankar places admit local uniformization in any characteristic

Abstract

We prove that every place P of an algebraic function field F | K of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field FP over K is equal to the transcendence degree of F | K , and the extension F P | K is separable. We generalize this result to the case where P dominates a regular local Nagata ring R ⊆ K of Krull dimension dim R ⩽ 2 , assuming that the valued field ( K , v P ) is defectless, the factor group v P F / v P K is torsion-free and the extension of residue fields F P | K P is separable. The results also include a form of monomialization.